超临界膜模型协方差的指数衰减
Exponential decay of covariances for the supercritical membrane model
摘要：
我们考虑的膜模型是$\mathbb Z^d$上的中心高斯场，其协方差阵由离散Bilaplacian的逆给出。我们施加一个$δ-$钉扎条件，给予奖励的强度$\varepsilon$的领域是$0$在任何网站的晶格。在本文中，我们证明了在尺寸$d\geq 5$协方差的钉扎字段衰减至少指数，而不是没有钉扎，其中的衰减是多项式的字段。证明是基于某些离散加权范数，渗流参数和伯努利支配结果的估计。
英文摘要：
We consider the membrane model, that is the centered Gaussian field on $\mathbb Z^d$ whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a $δ-$pinning condition, giving a reward of strength $\varepsilon$ for the field to be $0$ at any site of the lattice. In this paper we prove that in dimensions $d\geq 5$ covariances of the pinned field decay at least exponentially, as opposed to the field without pinning, where the decay is polynomial. The proof is based on estimates for certain discrete weighted norms, a percolation argument and on a Bernoulli domination result.
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